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Universality in Statistical Measures of Trajectories in Classical Billiard Systems

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Universality in Statistical Measures of Trajectories in Classical Billiard Systems Applied Mathematics, 2015, 6, 1407-1425 Published Online July 2015 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2015.68132 Universality in Statistical Measures of Trajectories in Classical Billiard Systems Jean-François Laprise, Ahmad Hosseinizadeh, Helmut Kröger Department of Physics, Laval University, Québec, Canada Email: [email protected], [email protected], [email protected] Received 13 January 2015; accepted 27 July 2015; published 30 July 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract For classical billiards, we suggest that a matrix of action or length of trajectories in conjunction with statistical measures, level spacing distribution and spectral rigidity, can be used to distin- guish chaotic from integrable systems. As examples of 2D chaotic billiards, we considered the Bu- nimovich stadium billiard and the Sinai billiard. In the level spacing distribution and spectral ri- gidity, we found GOE behaviour consistent with predictions from random matrix theory. We stu- died transport properties and computed a diffusion coefficient. For the Sinai billiard, we found normal diffusion, while the stadium billiard showed anomalous diffusion behaviour. As example of a 2D integrable billiard, we considered the rectangular billiard. We found very rigid behaviour with strongly correlated spectra similar to a Dirac comb. These findings present numerical evi- dence for universality in level spacing fluctuations to hold in classically integrable systems and in classically fully chaotic systems. Keywords Classical Chaos, Dynamical Billiards, Random Matrix Theory, Level Spacing Fluctuations, Universality 1. Introduction The idea to model apparently disordered spectra, like those of heavy nuclei, using random matrices was sug- gested in the mid-50’s by Wigner, and then formalized in the early 60’s in the work of Dyson and Mehta [1]-[6]. They showed that random matrices of Gaussian orthogonal ensembles (GOE) generate a Wigner-type nearest- neighbour level spacing (NNS) distribution [6]. In a seminal paper, Bohigas, Giannoni and Schmit (BGS) for- mulated a conjecture [7] stating that time-reversal invariant quantum systems with classically fully chaotic (ergodic) counterpart have universality properties given by random matrix theory (RMT). Experiments in nu- clear physics, for example, have shown that spectra originating from different heavy nuclei give the same How to cite this paper: Laprise, J.F., Hosseinizadeh, A. and Kröger, H. (2015) Universality in Statistical Measures of Trajec- tories in Classical Billiard Systems. Applied Mathematics, 6, 1407-1425. http://dx.doi.org/10.4236/am.2015.68132 J. F. Laprise et al. Wignerian energy level spacing distribution [8]. Universality properties in quantum chaos of bound systems, i.e. quantum systems with a fully chaotic classical counterpart, have now been demonstrated in many experiments, computational models and in theoretical studies [6] [9]-[14]. Theoretical support of the BGS conjecture came from the semiclassical theory of spectral rigidity by Berry [15] [16], who showed that universal behaviour in the energy level statistics is due to long classical orbits. Sieber and Richter [17] investigated the role of correlations between classical orbits. The semiclassical theory has been further developed by Müller et al. [18] [19]. In ref. [19] they presented the “core of a proof” of the BGS conjecture, which, by proving arguments previously used by Berry [15], show that in the semi-classical limit the periodic classical orbits determine the universal fluctua- tions of quantum energy levels. Further refinements have been made by Keating and Müller [20]. The study of classical strongly chaotic systems (Anosov systems) has revealed that central limit theorems (CLT) hold [21]-[26]. This has been proven for the 2D periodic Lorentz gas with finite horizon. The first step of a proof was given by Bunimovich and Sinai [21] and was completed by Bunimovich, Sinai and Chernov [23]. At macroscopic times, such deterministic chaotic system converges to Brownian motion, i.e. behaves like a random system [21] [23] [24] [26]. This is also supported by the existence of an average diffusion coefficient [21] [23] [27], showing the diffusive character of such chaotic system. Do classical fully chaotic systems also exhibit universality properties? This question was addressed by Argaman et al. [28], who showed that there is universal behaviour in 2 points correlation functions of actions of periodic orbits of classically chaotic systems. As examples they considered the deformed cat-map and the baker-map. This has been elaborated further in a number of studies by Dittes et al. [29], Aurich and Sieber [30], Cohen et al. [31], Tanner [32], Sano [33], Primack and Smilansky [34], Sieber and Richter [17] and Smi- lansky and Verdene [35]. Argaman et al. started from the assumption that spectral fluctuations of chaotic quantum systems follow the predictions of RMT and they derived a universal expression for classical cor- relation functions of periodic orbits via Gutzwiller’s semi-classical trace formula. They concluded “The real challenge, though, is to find out whether these action correlations can be explained on a completely classical level”. An answer was proposed by Laprise et al. [36] [37] who found universal behaviour in classical 2D billiards by looking at fluctuations in spectra of classical action/length matrices from billiard trajectories. They showed that one could distinguish chaos from integrability in classical systems using RMT and an analogue of the BGS- conjecture. In particular, they considered the Limaçon/Robnik family of billiards, which interpolates between the chaotic cardiod billiard and the integrable circular billiard. For the cardioid billiard, the level spacing dis- tribution Ps( ) and spectral rigidity ∆3 (L) were found to be consistent with the GOE behaviour predicted by RMT. For the interpolating case close to the circle, the behaviour approached a Poissonian distribution. The circular billiard itself was found to be very rigid and strongly correlated and yielded Ps( ) ∝−δ ( s 1) . The jump in behaviour at the transition to the circle is associated with the corresponding change in the symmetry group. This article extends the results of reference [36] in the following directions: 1) We consider the 2D rec- tangular billiard as another example of an integrable billiard. Compared to the circular billiard, this billiard has lower symmetry (no group property). Nevertheless, it displays strong spectral correlation and rigidity like the circular billiard. 2) We present numerical studies for other chaotic 2D billiards: the Sinai-billiard and the Bunimovich stadium billiard. 3) In order to understand the observed universal behaviour in chaotic billiards, we present arguments linking such behaviour to CLTs, diffusive and random walk behaviour. In particular, we present a mathematically rigorous result on the distribution of length of trajectories. The answers we found can be summarized as follows: For the 2D Sinai billiard and the 2D Bunimovich stadium billiard the level spacing distribution Ps( ) and spectral rigidity ∆3 (L) are consistent with pre- dictions of RMT (GOE behaviour), i.e. show universal behaviour. This behaviour is statistically the same as the one observed in quantum chaos (obtained from energy level spacing distributions). The implication of these findings is that RMT not only represents well the statistical fluctuation properties of the energy spectrum of chaotic quantum systems, but also those of the length spectrum of chaotic classical systems. Moreover, sta- tistical fluctuations obtained from spectra of action/length matrices clearly distinguish chaotic from integrable systems. 2. Length and Action of Trajectories In classical systems, chaos information is encoded in trajectories. According to the Alekseev-Brudno theorem 1408 J. F. Laprise et al. [38] the temporal length t is related via the Kolmogorov-Sinai K entropy to the information It( ) in the segment of trajectory, limIt( ) t= K . (1) t→∞ This motivates us to look at the length of trajectories and its fluctuation properties. Let Lxt( ( ),, xt ( ) t) denote the Lagrangian of a system, let xttraj ( ) denote a solution (trajectory) of the Euler-Lagrange equations, traj traj with boundary points xin ≡ xt( in ) and xfi ≡ xt( fi ) . Let traj tfi Σ=S x = d tLxt ( ( ), xt ( ) , t) traj , (2) ∫tin xx= denote the action over xttraj ( ) and let traj sx= fi Λ=xsd traj (3) ∫sx= in xx= traj denote the length of the trajectory xt( ) . We choose a finite set of discrete points X= { xx12,,, xN } . For all traj pairs of boundary points xxij, ∈ X, we compute a classical trajectory xij, , connecting those points. We suggest the construction of an action matrix and a length matrix, Σ= traj ijSx i, j , Λ=Λtraj ijx i, j , (4) where i and j are respectively the indices of the final and initial boundary points of the trajectory. Both matrices are viable for statistical analysis of classical chaos. In the case of billiard systems, we consider trajectories where the billiard particle moves with constant velocity u and constant kinetic energy E. Then the action and the length matrix are essentially equivalent, E Σ= Λ . (5) iju ij Numerical Calculation of Length Matrices When solving for two boundary points xi and x j in a billiard the solution is generally not unique; the number of possible trajectories varies with the number Nreb of rebounds. We have thus classified trajectories-and corresponding length matrices-according to Nreb . 5 For Nreb = 15 the number of trajectories is quite large ( Ntraj ∼10 ) . In order to limit computational cost, we considered smaller subsets of length matrices, in the following way. For a given number of rebounds Nreb and a given pair of boundary points {xxij, } , we determined the starting angles αν (measured from the normal to the billiard wall at the boundary point) which corresponds to a trajectory ν . That search was carried out in the range −π 2 ≤≤α π 2 .
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